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        <ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#支持向量机"><span class="toc-text"> 支持向量机</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#松弛变量和惩罚函数"><span class="toc-text"> 松弛变量和惩罚函数</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#线性svm"><span class="toc-text"> 线性SVM</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#必须掌握的知识点"><span class="toc-text"> 必须掌握的知识点</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#对线性svm的建模"><span class="toc-text"> 对线性SVM的建模</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#确定目标函数优化对象"><span class="toc-text"> 确定目标函数&amp;&amp;优化对象</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#构建决策面方程"><span class="toc-text"> 构建决策面方程</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#两个难题"><span class="toc-text"> 两个难题</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#问题的解决思路"><span class="toc-text"> 问题的解决思路</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#获得约束条件"><span class="toc-text"> 获得约束条件</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#求目标函数"><span class="toc-text"> 求目标函数</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#对方程进行求解"><span class="toc-text"> 对方程进行求解</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#拉格朗日对偶"><span class="toc-text"> 拉格朗日对偶</span></a></li></ol></li></ol>
    
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        <h2 id="支持向量机"><a class="markdownIt-Anchor" href="#支持向量机"></a> 支持向量机</h2>
<ul>
<li>向量内积(点乘)，对应相乘相加</li>
<li>或者表示为<code>|x|*|y|*cosθ</code></li>
<li>范数，平方和开根号</li>
<li>推荐b站覃秉丰的机器学习</li>
<li>核函数就是为了简便内积的运算而引入的，因为有些时候非线性的数据集不好处理，要进行低维转化为高维来寻找超平面，由此引入了核函数</li>
</ul>
<p>支持向量机<code>（SVM）support vector machine</code>，主要是用来解决二分类问题的，他被认为是最好的分类方法。因为他能解决线性回归不能解决的分类问题</p>
<p><strong><code>SVM</code>就是试图把棍放在最佳位置，好让在棍的两边有尽可能大的间隙。这个间隙就是球到棍的距离。</strong></p>
<p><img src="https://img-blog.csdn.net/20170923170200381?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>即使放了更多的球，棍仍然是一个好的分界线。</p>
<p><img src="https://img-blog.csdn.net/20170923170228849?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>现在，假如我们现在有这样类型的数据，我们不能再使用一条直线将他们很好地区分开来了，使用<code>SVM</code>却可以。</p>
<p><img src="https://img-blog.csdn.net/20170923170253159?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>我们需要对二维平面进行升维处理，找到一个最好的平面，将数据分类，我们将这个平面称为<strong>超平面</strong></p>
<p><img src="https://img-blog.csdn.net/20170923170319898?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<h2 id="松弛变量和惩罚函数"><a class="markdownIt-Anchor" href="#松弛变量和惩罚函数"></a> 松弛变量和惩罚函数</h2>
<p>因为实际的数据不可能没有错误的**(非线性情况)**，总会有分类错误的点(或者苹果跑到梨子那边了)</p>
<ul>
<li>松弛变量就是让支持向量(曲线)不断移动</li>
<li>惩罚函数就是让分错的点越少越好(最好的时候就是刚好与分错的点相切)</li>
</ul>
<h2 id="线性svm"><a class="markdownIt-Anchor" href="#线性svm"></a> 线性<code>SVM</code></h2>
<p>现在先看一下平面二分类的问题</p>
<p><img src="https://img-blog.csdn.net/20170923170544874?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>黑色实线为分界线，术语称为“决策面”，每个决策面对应了一个线性分类器，虽然从分类结果上看，分类器A和分类器B的效果是相同的。但是他们的性能是有差距的。<strong>在”决策面”不变的情况下，我又添加了一个红点。可以看到，分类器B依然能很好的分类结果，而分类器C则出现了分类错误。</strong></li>
</ul>
<p><img src="https://img-blog.csdn.net/20170923170619352?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li><strong>而这个真正的最优解对应的两侧虚线所穿过的样本点，就是<code>SVM</code>中的支持样本点，称为”支持向量”。</strong></li>
</ul>
<h2 id="必须掌握的知识点"><a class="markdownIt-Anchor" href="#必须掌握的知识点"></a> 必须掌握的知识点</h2>
<ol>
<li>支持向量机的优点，即使数据再大，也比较好用，因为只用关心离决策面附近的几个点就行了</li>
<li>核函数，将低维映射到高维的一种方法，但是高维的运算(内积)运算量是非常大的，核函数运算厉害之处是，即使映射到高维，但是数据处理依旧是在低维中进行</li>
<li>线性核函数可以直接使用，不需要设置参数，径向基核函数可以将低维映射到无穷维，但是要设置一些参数，使用广泛的原因是因为，就算参数设置的不是很完美，依旧可以得到比较好的结果</li>
<li><code>SVM</code>算法优点是：计算量不大，只用计算几个决策边界附近的点就行，错误率比较低</li>
<li>缺点是：对调制的参数比较敏感，而且只适用于二分类问题</li>
</ol>
<h2 id="对线性svm的建模"><a class="markdownIt-Anchor" href="#对线性svm的建模"></a> 对线性<code>SVM</code>的建模</h2>
<h3 id="确定目标函数优化对象"><a class="markdownIt-Anchor" href="#确定目标函数优化对象"></a> 确定目标函数&amp;&amp;优化对象</h3>
<ul>
<li>
<p>目标函数，就是我们的目的是什么，在上面的例子中，我们希望将支持向量到决策面的距离越大，也就是<strong>分类间隔</strong></p>
</li>
<li>
<p>优化对象，就是我们要调整什么参数，才能使得这个目标对象达到最优，这里我们指的是决策面</p>
</li>
<li>
<p>为了解决实际问题，我们都需要进行公式表示，来对实际数据进行建模，并用数学来对问题进行解答，在建模的过程中，我们通常需要对实际问题进行假设</p>
</li>
</ul>
<h3 id="构建决策面方程"><a class="markdownIt-Anchor" href="#构建决策面方程"></a> 构建决策面方程</h3>
<p>我们都知道二维空间下一条直线的方式如下所示：</p>
<p><img src="https://img-blog.csdn.net/20170923170727730?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>现在我们做个小小的改变，让原来的<code>x</code>轴变成<code>x1</code>，<code>y</code>轴变成<code>x2</code>。</p>
<p><img src="https://img-blog.csdn.net/20170923170755765?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>移项得：</p>
<p><img src="https://img-blog.csdn.net/20170923170822756?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>将公式向量化得：</p>
<p><img src="https://img-blog.csdn.net/20170923170855067?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>进一步向量化，用w列向量和x列向量和标量γ进一步向量化：</p>
<p><img src="https://img-blog.csdn.net/20170923170921357?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>其中，向量w和x分别为：</p>
<p><img src="https://img-blog.csdn.net/20170923170949079?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>二维空间的直线方程已经推导完成，将其推广到n为空间，就变成了超平面方程。(一个超平面，在二维空间的例子就是一个直线)但是它的公式没变，依然是：</p>
<p><img src="https://img-blog.csdn.net/20170923171351900?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>不同之处在于：</p>
<p><img src="https://img-blog.csdn.net/20170923171420742?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>我们已经顺利推导出了”决策面”方程，它就是我们的超平面方程，之后，我们统称其为超平面方程。</p>
<p><img src="https://img-blog.csdn.net/20170923171540791?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>这个图非常重要，为了求出图中的距离d，我们只需找到离直线最近的点，再算出点到直线的距离即可</li>
</ul>
<p><img src="https://img-blog.csdn.net/20170923171610356?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>现在，将直线方程扩展到多维，求得我们现在的超平面方程，对公式进行如下变形：</p>
<p><img src="https://img-blog.csdn.net/20170923171637926?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>这个d就是”分类间隔”。其中<code>||w||</code>表示w的二范数，求所有元素的平方和，然后再开方。比如对于二维平面：</p>
<p><img src="https://img-blog.csdn.net/20170923171706926?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>那么，</p>
<p><img src="https://img-blog.csdn.net/20170923171733417?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>推广到多维不难理解，<strong>范数就是变量系数平方和再开根号</strong></li>
<li>接下来要解决的就是求距离最大值了</li>
</ul>
<h2 id="两个难题"><a class="markdownIt-Anchor" href="#两个难题"></a> 两个难题</h2>
<ul>
<li>如何判断决策面能够比较好地将两类分开完全呢</li>
<li>如何找到距离决策面最近地两点呢</li>
</ul>
<p>也就是说，为了能够实现上面地分类，我们必须先找到一条直线，能够对数据进行正确地分类，这是最基本地要求，然后在已经分类好地条件下，找到距离决策面最近地几个点，调整决策面位置，使他们的距离到最大。(找到最优解)</p>
<h2 id="问题的解决思路"><a class="markdownIt-Anchor" href="#问题的解决思路"></a> 问题的解决思路</h2>
<p>这个二维平面上有两种点，我们分别对它们进行标记：</p>
<ul>
<li>红颜色的圆点标记为1，我们人为规定其为正样本；</li>
<li>蓝颜色的五角星标记为-1，我们人为规定其为负样本。</li>
</ul>
<h3 id="获得约束条件"><a class="markdownIt-Anchor" href="#获得约束条件"></a> 获得约束条件</h3>
<p>对每个样本点xi加上一个类别标签yi：</p>
<p><img src="https://img-blog.csdn.net/20170923171905897?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li><strong>首先，我们要对不同样本点进行标签分类，为了后面方便计算，我们在这里将数据分成-1和1两类</strong></li>
</ul>
<p><img src="https://img-blog.csdn.net/20170923171933138?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li><strong>平面上的所有点肯定是满足这个不等式的，即-1类的点，肯定在决策面的一边</strong></li>
</ul>
<p><strong>现在将条件变得更苛刻一些，即点到直线的距离，都是大于间隔<code>d</code>的</strong></p>
<p><img src="https://img-blog.csdn.net/20170923172000544?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>式两边都除以d，就可以得到：</p>
<p><img src="https://img-blog.csdn.net/20170923172027169?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>其中：</p>
<p><img src="https://img-blog.csdn.net/20170923172051319?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>让我们对<code>wd</code>和<code>γd</code>重新起个名字，就叫它们<code>w</code>和<code>γ</code>。因此，我们就可以说：”对于存在分类间隔的两类样本点，我们一定可以找到一些超平面面，使其对于所有的样本点均满足下面的条件：”</p>
<p><img src="https://img-blog.csdn.net/20171030230804176?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>这个就是我们的约束条件，就是将点区分开来需要满足的条件，也就是说，如果我找到了一个决策面，不满足这个条件，那么这个决策面就是错误的</li>
</ul>
<p>进一步地，我们将约束条件写成：</p>
<p><img src="https://img-blog.csdn.net/20170923172259072?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>因为<code>yi</code>只能取1或者-1，而由上面那个公式知道，负数乘负数依旧是正数</li>
</ul>
<h2 id="求目标函数"><a class="markdownIt-Anchor" href="#求目标函数"></a> 求目标函数</h2>
<p>前面我们知道目标函数，也就是关于距离<code>d</code>的方程</p>
<p><img src="https://img-blog.csdn.net/20170923172358421?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p><strong>为了方便计算，我们先假设：任意支持向量上的点，到决策面的距离满足这条件</strong></p>
<p><img src="https://img-blog.csdn.net/20170923172456094?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>进而将求d最大值问题化成了</p>
<p><img src="https://img-blog.csdn.net/20170923172524509?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>要求d的最大值，转换成求w的最小值</li>
</ul>
<p>因为，我们只关心支持向量上的点。随后我们求解d的最大化问题变成了||w||的最小化问题。进而||w||的最小化问题等效于</p>
<p><img src="https://img-blog.csdn.net/20170923172617826?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>这样表示是为了后面好求导，而且w的极值和该函数取极值的点是一样的（平方不影响）</li>
</ul>
<p>我们将最终的目标函数和约束条件放在一起进行描述：</p>
<p><img src="https://img-blog.csdn.net/20170923172648006?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<blockquote>
<p>缩写<code>s.t.</code>表示<code>”Subject to”，</code>是”服从某某条件”的意思。</p>
</blockquote>
<h2 id="对方程进行求解"><a class="markdownIt-Anchor" href="#对方程进行求解"></a> 对方程进行求解</h2>
<p>为了对方程进行求解，我们需要先了解下拉格朗日函数</p>
<p><img src="https://img-blog.csdn.net/20170923173128075?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>满足这个条件的，都可以用拉格朗日函数求极值</li>
</ul>
<p><img src="https://img-blog.csdn.net/20170923172648006?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<ul>
<li>我们只需将上面第二个等式移项一下即可(左边移到右边)</li>
</ul>
<h3 id="拉格朗日对偶"><a class="markdownIt-Anchor" href="#拉格朗日对偶"></a> 拉格朗日对偶</h3>
<p>大概意思就是说，我们可以看到，要求下面函数的最小值，依旧比较困难，我们可以利用拉格朗日对偶将问题转换(最大值的最小值，转换成求最小值的最大值)</p>
<p><img src="https://img-blog.csdn.net/20170923172648006?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>接着利用对应的数学公式，我们可以将问题转变成：</p>
<p><img src="https://img-blog.csdn.net/20170923174148322?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>
<p>首先固定α，要让L(w,b,α)关于w和b最小化，我们分别对w和b偏导数，令其等于0，即：</p>
<p><img src="https://img-blog.csdn.net/20170923174216260?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvYzQwNjQ5NTc2Mg==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="img" /></p>

      
       
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<div class="article_copyright">
    <p><span class="copy-title">文章标题:</span>支持向量机</p>
    <p><span class="copy-title">文章字数:</span><span class="post-count">2.3k</span></p>
    <p><span class="copy-title">本文作者:</span><a  title="Miki Zhu">Miki Zhu</a></p>
    <p><span class="copy-title">发布时间:</span>2020-03-13, 20:53:20</p>
    <p><span class="copy-title">最后更新:</span>2020-03-14, 19:48:32</p>
    <span class="copy-title">原始链接:</span><a class="post-url" href="/2020/03/13/%E6%94%AF%E6%8C%81%E5%90%91%E9%87%8F%E6%9C%BA%E7%AE%97%E6%B3%95/" title="支持向量机">http://mikiblog.online/2020/03/13/%E6%94%AF%E6%8C%81%E5%90%91%E9%87%8F%E6%9C%BA%E7%AE%97%E6%B3%95/</a>
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        <span class="copy-title">版权声明:</span><i class="fa fa-creative-commons"></i> <a rel="license noopener" href="http://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank" title="CC BY-NC-SA 4.0 International" target = "_blank">"署名-非商用-相同方式共享 4.0"</a> 转载请保留原文链接及作者。
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